The U.S. has agreed to give up supervision of the Internet policy-making body that controls domain names, hoping to satisfy countries that want more international control over the Internet. This week, Washington will find out if its actions have eased global tensions over its cyberspying activities.

World representatives are arriving in Brazil for Net Mundial, a two-day meeting about Internet governance that starts Wednesday in São Paulo. The meeting comes amid fallout over spying by the U.S. National Security Agency that...